Is Matching Dimension Enough for Linear Transformation Invertibility?

[Bipolarity]

For a linear transformation to be invertible, is it a requirement that the domain and codomain be the same vector space, or merely that they have the same dimension? My intuition tells me they merely need the same dimension but someone can correct me please?

BiP

 

[Office_Shredder]

You have to be a bit careful - the range of the linear transformation needs to have the same dimension, not just the codomain. But there is no requirement they be exactly the same vector space (in fact two vector spaces which are the same dimension are always isomorphic, so in a sense they are the same vector space as far as any linear algebra operations go)

 

[mathwonk]

your intuition is correct. two vector spaces are isomorphic precisely when they have the same dimension.

 

[HallsofIvy]

For example, the function f(a, b)= a+ bx is an invertible linear transformation for R², the vector space of ordered pairs of real numbers, to P², the vector space of linear polynomials, both of which have dimension 2.